On the Reverse Inequality of Riesz transform on metric cone with potential

Let $M=(0,\mathrm{\infty }{)}_r\times Y$ be a $d$-dimensional ($d\ge 3$) metric cone with metric<br/>$g=d{r}^2+r^{2}h$, where $(Y,h)$ is a closed Riemannian manifold. Let<br/>$H=\mathrm{\Delta }+V_0\mathrm{/}{r}^2$ be the associated Schrodinger operator, with<br/>$V_0\in C^{\mathrm{\infty }}(Y)$ satisfying the positivity condition<br/>${\mathrm{\Delta }}_Y+V_0+(d-2{)}^2\mathrm{/}4>0$. First, we complement previous results by proving<br/>Lorentz-type endpoint estimates for the Riesz transform $\mathrm{\nabla }{H}^{-1\mathrm{/}2}$:<br/>it is of restricted weak type at both endpoints of its $L^p$-boundedness range.<br/>Second, we establish the sharp reverse inequality<br/>$\mathrm{\parallel }{H}^{1\mathrm{/}2}f{\mathrm{\parallel }}_{L^p}\le C(\mathrm{\parallel }\mathrm{\nabla }f{\mathrm{\parallel }}_{L^p}+\mathrm{\parallel }f\mathrm{/}r{\mathrm{\parallel }}_{L^p})$<br/>which holds if and only if<br/>$$<br\mathrm{/}>\frac{d}{\min ((d+4)\mathrm{/}2+{\mu }_0,d)}<br\mathrm{/}><p<<br\mathrm{/}>\frac{d}{\max ((d-2)\mathrm{/}2-{\mu }_0,0)}\mathrm{.}$$